On the invariant uniform Roe algebra

Abstract

Let $\Gamma$ be a countable discrete group. We show that $\Gamma$ has the approximation property if and only if $\Gamma$ is exact and for any operator space $S \subseteq \K(H)$ we have $\Cu(Γ)^Γ \otimes S = (\Cu(Γ) \otimes S)^Γ$, where $\Cu(Γ)$ is the uniform Roe algebra with the right adjoint $\Gamma$-action. This answers a question of J. Zacharias. We also show that characterisations of several properties of $\Gamma$ in terms of the reduced group \cast-algebra apply to the invariant uniform Roe algebra $\Cu(Γ)^Γ$.

6 pages